Borel-fixed ideals and reduction number

The aim of this paper is to study the relationship between reduction numbers and Borel-fixed ideals in all characteristics. By definition, Borel-fixed ideals are closed under certain specializations which is similar to the strong stability. We will estimate the number of monomials which can be specialized to a given monomial. As a consequence, we obtain a combinatorial version of the well-known Eakin-Sathaye's theorem which bounds the reduction number in terms of the Hilbert function. Furthermore, we show that the bound of Eakin-Sathaye's theorem is attained by the reduction number of a lex-segment monomial ideal. This result answers a question of Conca in the affirmative. We will also show that the reduction number of the lex-segment ideal is bounded exponentially by the reduction number of the given ideal.Comment: 11 page

Characterization of an Orthotopic Rat Model of Glioblastoma Using Multiparametric Magnetic Resonance Imaging and Bioluminescence Imaging

Glioblastoma multiforme (GBM) is a lethal and incurable disease. The C6 rat model of GBM shares several similarities to human GBM and longitudinal non-invasive imaging may allow tumour features to be studied. In this thesis, a multimodality imaging framework, consisting of bioluminescence imaging (BLI) and multiparametric magnetic resonance imaging (mpMRI), was applied to the C6 rat model to characterize the growth of orthotopic tumours. BLI signal, a measure of cell viability, tended to increase and then decrease in the majority of animals, whereas tumour volume (from MRI) continually increased. Cellular viability and tumour volume did not correlate across all days, highlighting the value of using complimentary imaging modalities. Apparent diffusion coefficient maps and immunohistochemistry suggests decreases in BLI signal are in part due to decreased tumour cellularity (i.e. necrosis). This is the first use of BLI and mpMRI to characterize this model, and highlights the inter-subject variability in tumour growth